| Summary: | In this thesis we study the birational geometry of the blow-ups of toric varieties of Picard number one at a general point, and the blow-up of projective spaces along points and lines. In Chapter 2 we consider the blow-up X of P^3 at 6 points in very general position and the 15 lines through the 6 points. We construct an infinite-order pseudo-automorphism \phi_X on X, induced by the complete linear system of a divisor of degree 13. The effective cone of X has infinitely many extremal rays and hence, X is not a Mori Dream Space. The threefold X has a unique anticanonical section which is a Jacobian K3 Kummer surface S of Picard number 17. The restriction of \phi_X on S realizes one of Keum's 192 infinite-order automorphisms of Jacobian K3 Kummer surfaces. In general, we show the blow-up of P^n (n\geq 3) at (n+3) very general points and certain 9 lines through them is not Mori Dream, with infinitely many extremal effective divisors. As an application, for n\geq 7, the blow-up of \bar{M}{0,n} at a very general point has infinitely many extremal effective divisors. In Chapter 3 we show that the blow-up of the 7-th Losev-Manin moduli space is log-Fano. We review the known extremal effective divisors on \bar{M}{0,n} and their symmetrized images in the Losev-Manin moduli space. In Chapter 4 and 5 we study the blow-ups of toric varieties of Picard number one at the identity point of the torus. We study whether such blow-ups are Mori Dream Spaces. Chapter 4 shows that for some of these toric surfaces, the question whether the blow-up is a Mori Dream Space is equivalent to countably many planar interpolation problems. We state a conjecture which generalizes a theorem of Gonzalez and Karu. We give new examples of Mori Dream Spaces and not Mori Dream Spaces among these blow-ups. Chapter 5 describes a sufficient condition in higher dimensions for the blow-up of a weighted projective space P(a,b,c,d_1,\cdots,d_{n-2}) at the identity point not to be a Mori Dream Space. We exhibit several infinite sequences of weights satisfying this condition in all dimensions n\geq 3--Author's abstract
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